Different Ways of Hydrogen Bonding in Water - Why Does Warm Water

Article pubs.acs.org/JCTC

Different Ways of Hydrogen Bonding in Water - Why Does Warm Water Freeze Faster than Cold Water? Yunwen Tao,† Wenli Zou,† Junteng Jia,‡ Wei Li,‡ and Dieter Cremer*,† †

Computational and Theoretical Chemistry Group (CATCO), Department of Chemistry, Southern Methodist University, 3215 Daniel Avenue, Dallas, Texas 75275-0314, United States ‡ Institute of Theoretical and Computational Chemistry, Key Laboratory of Mesoscopic Chemistry of MOE, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing 210023, P. R. China S Supporting Information *

ABSTRACT: The properties of liquid water are intimately related to the H-bond network among the individual water molecules. Utilizing vibrational spectroscopy and modeling water with DFT-optimized water clusters (6-mers and 50-mers), 16 out of a possible 36 different types of H-bonds are identified and ordered according to their intrinsic strength. The strongest H-bonds are obtained as a result of a concerted push−pull effect of four peripheral water molecules, which polarize the electron density in a way that supports charge transfer and partial covalent character of the targeted H-bond. For water molecules with tetra- and pentacoordinated O atoms, H-bonding is often associated with a geometrically unfavorable positioning of the acceptor lone pair and donor σ*(OH) orbitals so that electrostatic rather than covalent interactions increasingly dominate H-bonding. There is a striking linear dependence between the intrinsic strength of H-bonding as measured by the local H-bond stretching force constant and the delocalization energy associated with charge transfer. Molecular dynamics simulations for 1000-mers reveal that with increasing temperature weak, preferentially electrostatic H-bonds are broken, whereas the number of strong H-bonds increases. An explanation for the question why warm water freezes faster than cold water is given on a molecular basis.

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workers using second order perturbation theory.38 DFT benchmark calculations utilizing B3LYP, X3LYP, and M06type of XC-functionals for predicting binding energies of water clusters up to 20 molecules have been carried out by Bryantsev and co-workers.39 Parthasarathi and co-workers found that linear chains of up to 20 water molecules lead to dipole moments as high as 41 D thus emphasizing the cooperative effect of H-bonding in larger clusters.40 An interesting study on the polarizability of water clusters and the charge flow through H-bonds in the presence of internal and external electric fields was carried out by Yang and co-workers.41 The electron density at the critical points of the H-bonds of a water cluster was analyzed by Neela and co-workers who predicted an increase of the density with the cluster size.51 Iwata pointed out the importance of charge transfer and dispersion energies in (H2O)20 and (H2O)25, which he found to depend on the O···O distance.42 Lenz and co-workers43 calculated the vibrational spectra of water clusters containing up to 30 water molecules. They found a correlation between the red-shift of the O−H donor stretching frequency and the type of H-bond based on the coordination numbers of the O atoms being involved. The importance of the collective electrostatic effects on H-bonding

INTRODUCTION The understanding of hydrogen bonding (H-bonding) is essential for unravelling many biological and environmental phenomena.1−6 H-bonding dominates the noncovalent interactions between the molecules in liquid water, and in this way H-bonding is ultimately responsible for the unique properties of water. Essential for the understanding of the complex structure and dynamics of liquid water7 is the study of Hbonding with the help of quantum chemical methods. If an atomistic approach is used, liquid water can be modeled by using clusters of water molecules. The smallest of such clusters, the water dimer, is only used for reference purposes, and its properties in connection with H-bonding are fairly wellknown.8−20 Also, larger clusters with three to six water molecules have been reliably described and have helped to extend the understanding of H-bonding between water molecules.21−36 Less frequent are high-accuracy investigations of larger water clusters.37 Most of these investigations have been carried out at the Hartree−Fock (HF), Density Functional Theory (DFT), or perturbation theory level. For example, the investigation of 20-mers (clusters with 20 water molecules),38−41 25-mers,42 30-mers up to 40-mers,43−49 or even 60-mers has to be reported.50 Noteworthy in this connection is that the vibrational spectra of 20-mers have been investigated in detail by Xantheas and co© 2016 American Chemical Society

Received: July 23, 2016 Published: December 6, 2016 55

DOI: 10.1021/acs.jctc.6b00735 J. Chem. Theory Comput. 2017, 13, 55−76

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Journal of Chemical Theory and Computation

to the phenomenon that warm water freezes faster than cold water. The chemical relevance and the conclusions of the current investigation will be summarized in the last section.

as caused by the nonimmediate environment in liquid water models was emphasized by Bako and co-workers.44 Qian and co-workers did systematic studies on water clusters of different size ranging from the dimer to 34-mers using HF/6-31G(d).47 An attempt was made by Huang and co-workers to predict the far-infrared spectra of water clusters up to 38 molecules with DFT and to relate them to observed THz spectra.48 Specific forms of water clusters were investigated by various authors (spiro-cyclic,45 fullerene-shaped50). Frogato and co-workers performed ab initio Born−Oppenheimer molecular dynamics (MD) simulations for 69-mer clusters containing an excess electron.52 Clusters with up to 280 water molecules were investigated by Loboda and co-workers who determined averaged H-bond energies.53 Turi used mixed quantum-classical MD simulations for a cluster consisting 1,000 water molecules either in neutral state or with an excess electron.54 A quantum simulation of water was carried out by Wang and co-workers.55 In these investigations, the intrinsic strength of the H-bond in water clusters or liquid water could not be determined. Instead one attempted to obtain indirect evidence by analyzing binding energies, H-bond distances, vibrational frequencies, electron densities at the H-bond critical point, and other molecular properties, which can only provide a qualitative measure of the H-bond strength as they relate to all intermolecular forces and interaction energies. In this work, we present the harmonic vibrational frequencies of water clusters containing 50 molecules (50-mers) that can be considered as suitable models for distinguishing between different H-bond types. For the first time, we will provide a detailed account on H-bonding in water clusters, which can be considered as suitable models for liquid water. In connection with this general goal, we pursue the following objectives: (i) We will investigate how many of the 36 possible standard Hbond types (excluding pentacoordination of oxygen and Hbond bifurcation) are needed to analyze H-bonding in the 50mers. (ii) We will characterize the various H-bonds according to the intrinsic strength of their interactions, which we will characterize with the properties of the H-bond stretching vibrations using the theory of Konkoli and Cremer for analyzing local vibrational modes.56−59 For this purpose, we will derive a H-bond strength order (BSO) value, which will provide a quantitative measure to compare different H-bonds in the water clusters investigated. (iii) H-bonding results from the noncovalent interactions of a H-bond donor (D) and a H-bond acceptor (A). Accordingly, we will investigate to which extent the properties of D and those of A are varied by H-bonding. Are there relationships between H-bond stretching force constants, covalent and electrostatic bond character, electron and energy density properties at the H-bond critical points, or the H-bond lengths? (iv) Is there a relationship between the strength of the OH donor bond and that of the H-bond, which can be used to characterize the latter via properties of the former?5,6 (v) Finally, we will make an attempt to relate the structure of a water-cluster to the macroscopic properties of liquid water by utilizing MD simulations of 1000-mers. In this connection, we will investigate the question why warmer water freezes more quickly than colder water.60−66 The results of this investigation will be presented in the following order. In Section 2, we will describe the computational methods used in this work. In Section 3, the different Hbonds of the 50-mers will be analyzed, and a suitable way of describing them will be worked out. The results of this analysis will be applied in Section 4 to provide a molecular explanation

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COMPUTATIONAL METHODS Equilibrium geometries and normal vibrational modes were calculated using the ωB97X-D density functional,67,68 which was chosen because it provides a reliable description of noncovalent interactions in cases where dispersion and other long-range van der Waals interactions play an important role.69−72 Pople’s triple-ζ basis 6-311G(d,p) was augmented by diffuse functions for H and O atoms. The 6-311++G(d,p) basis set73−75 thus obtained contained 1800 basis functions for the 50-mers. The calculations of the normal mode vectors and frequencies were carried out with the GEBF (generalized energy-based fragmentation) method76,77 at the ωB97X-D/6311++G(d,p) level. The analytical gradient of the GEBF method77 was used for the geometry optimization, whereas for the GEBF Hessian an approximate expression was employed.78 The usefulness of GEBF-ωB97X-D in the case of the water clusters was first tested by carrying out calculations for 20-mers and comparing results obtained with CCSD(T)79 in the form of GEBF-CCSD(T). GEBF-ωB97X-D turned out to be both reliable and cost efficient. In the GEBF-ωB97X-D calculations, each water molecule was selected as a fragment, and the distance threshold was set to 4.0 Å, i.e. at least one atom is within this limit. The maximum number of fragments in each subsystem was limited to seven. Natural population analysis (NPA)80,81 charges were employed as background charges, and two-fragment subsystems with a distance threshold of 8.0 Å were considered for corrections. The DFT calculations were carried out using a pruned (75,302) fine grid82,83 and tight convergence criteria in the geometry optimizations to guarantee a reliable calculation of vibrational frequencies. The initial geometries of the 50-mers were taken from MD calculations using a TIP4P force field.84 The optimized geometries are given in the Supporting Information (SI). The relative energies of the complexes used in this investigation are 0.0 kcal/mol (cluster A; absolute energy: −3822.655729 hartree), −0.62 (B), 10.30 (C), and 4.40 kcal/mol (D). The lowest vibrational frequencies obtained in this way are 23.7, 26.3 (cluster A); 25.2, 29.6 (B); 18.0, 21.3 (C); 17.9, 23.4 cm−1 (D). Another water complex leading to an imaginary frequency was excluded from the investigation. Electron density and energy density distributions were calculated using ωB97X-D rather than GEBF-ωB97X-D. The charge transfer analysis was carried out on the basis of calculated NPA charges.80,81 A topological analysis of the electron density distribution ρ(r) was performed.85 The nature of the H-bond was determined by the energy density H(r) calculated at the H-bond critical point rb and the application of the Cremer−Kraka criteria for covalent bonding: (i) A zero-flux surface and bond critical point rb have to exist between the atoms in question (necessary condition). (ii) The local energy density at H(rb) must be negative and thereby stabilizing (sufficient condition for covalent bonding). A positive H(rb) indicates a dominance of electrostatic interactions.86−88 Hence, the Cremer−Kraka criteria can reveal whether H-bonding is covalent, electrostatic, or a mixture of both (values close to zero). The covalent character of the H-bond was estimated by calculating the delocalization energy ΔE(del), which is associated with the charge transfer from a lone pair orbital of 56


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Figure 1. Coding of the 16 H-bond types discussed for the 50-mers by the notation DcD(ia,jd) − AcA(ka,ld). The integers ia, jd, ka, and ld give the peripheral (external) H-bonds directly embedding the targeted H-bond, i.e. the acceptor (a) and donor (d) H-bonds of D and A, which are ≤2 for the D or A water molecule. The superscripts cD and cA are the coordination numbers of O(D) and O(A), respectively, which vary between 3 and 4 and must be distinguished from the number of H-bonds m(D) and m(A). In black bold print the average m(AD) of m(D) and m(A) is given. For the strongest H-bond D4(20) − A3(02) (in short: 20-02) red arrows indicate the direction of charge polarization, which supports the covalent character of this H-bond. For other H-bonds, the optimal 20-02 arrangement is perturbed as indicated for the D4(21) − A4(11) (in short: 21-11) H-bond by dashed red arrows. For each type of H-bond, the deviation is given in blue as in the case of the 21-12 H-bond: |21-11 - 20-02 | = 01-11. The distortion descriptor (see text) is given as a negative blue number in the upper right of each drawing where an encircled number defines the position of the H-bond in a strength order from 1 (strongest H-bond) to 16 (weakest H-bond) according to the average BSO values of Figure 3. The 10-01 bond value has been added as reference (BSO: n = 0.399; see text). On the left, the distortion relationship between rows of the matrix of H-bond types is given in brown color.

= 1, ···, Nvib with Nvib = 3N − L, N: number of atoms; L: number of translations and rotations), G is the Wilson matrix for the kinetic energy,90 and Λ is a diagonal matrix containing the vibrational eigenvalues λμ = 4π2c2ω2μ. In the expression for the eigenvalues, ωμ represents the vibrational frequency of mode dμ. Matrix Γ in eq 2 is the inverse force constant matrix, which is usually called compliance matrix.92 Matrix R̃ corresponds to D̃ in the local mass-weighted formulation (indicated by the tilde)91

A to an antibonding OH orbital of D thus leading to an increase of the electron density in the range of the H-bond. The magnitude of ΔE(del) was determined by second order perturbation theory81 where both lp(O) → σ*(OH)-contributions to ΔE(del) for a given O···OH-interaction were included (see Section 3). The intrinsic strength of the H-bond was determined by using the local H-bond stretching force constant.59,89 The local vibrational modes of Konkoli and Cremer56 are based on the solution of the Wilson equation of vibrational spectroscopy90 FqD = G−1DΛ

R̃ = FqD̃

(1)

and the partitioning is into diagonal (d) and off-diagonal (od) parts. The parameter λ controls kinematic (mass) coupling, i.e. for λ = 0 the local description is obtained and for λ = 1 the Wilson equation reformulated in terms of compliance matrix and R modes. Solution of the Wilson equation requires the diagonalization of matrix Fq to give the matrix K. In this way, the electronic

in the form91 (Gd + λGod)R̃λ = (Γqd + λ Γqod)R̃λΛλ

(3)

(2)

In these two equations, Fq is the calculated force constant matrix expressed in internal coordinates qn, D collects the corresponding vibrational eigenvectors dμ as column vectors (μ 57


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Journal of Chemical Theory and Computation coupling between the local modes is eliminated. Solving the mass-decoupled Wilson equation leads to the mass-decoupled local modes an,91 which can be written as56,58,93

an =

periodic clusters were taken out using equal time intervals of 1 ps (picosecond). For the analyses, a H-bond was considered to be given if the H···O distance is between 1.5 and 2.2 Å, and the O−H···O angle is larger than 100.0°. PBC were also employed for determining H-bonds. In the analysis of the (H2O)50 clusters, the 2.2 Å limit turned out to be a reasonable cutoff value. There are 6 × 6 = 36 standard H-bond types (excluding bifurcated Hbonds and pentacoordination at the O atom, i.e. the maximum number of H-bonds per water molecule is limited to 4). If pentacoordination is included another 36 types of H-bonds are possible, whereas bifurcation adds another 71 H-bond types (maximum number of H-bonds ≤5) thus leading to a total of 11 × 13 = 143 different H-bond types. For the bifurcated Hbonds, the H atom of the O−H donor bond is within 2.4 Å with regard to two neighboring O atoms.107 In the equilibrium structures of the 50-mers, only a fraction of the maximally possible standard H-bonds can occur. These are shown in Figure 1 where the H-bond of the cyclic water hexamer, (H2O)6, is used as a suitable reference (see the SI for the equilibrium geometry). The average number of H-bonds per water molecule was obtained by the formula mav = 2 × N(H bonds)/N(water molecules) (N: number). In addition, we use the quantities m(D) and m(A), which give the total number of H-bonds of the D- and the A-water for a given type of H-bond (Figure 1). Accordingly, the average m(AD) is equal to [m(D) + m(A)]/2. Similarly, the number of peripheral H-bonds can be given by mp(D), mp(A), etc. Parameters m and mp are associated with the dimer D−A and do not consider peripheral H2O molecules (mav varies only slightly from 1.5 to 1.75 for the cases considered and therefore is less useful for the characterization of the types of H-bonds shown in Figure 1). Beside calculating the charge transfer between the interacting monomers using NPA values, we also calculated the difference density distribution Δρ(r) = ρ(Complex,r) − ρ(Monomer1,r) − ρ(Monomer2,r), which was determined and plotted for the complex-enveloping surface of an electron density distribution of 0.001 e/Bohr3. For the situation of six water molecules as in 20-02 (see below), the geometry was taken from one of the 2002 H-bonds of cluster A. The polarization effect caused by the four peripheral H2O was determined by subtracting from the hexamer density that of the trimers on the D and A side as well as that of the central D−A dimer and then adding the density of the D and A water monomer (short notation: 6-2 × 3-2 + 1 + 1) all calculated in the geometry of the hexamer. In this way, the density contributions of the trimers and the difference density of the central dimer were eliminated so that the push− pull effect of the peripheral water molecules on the central dimer unit becomes visible. For the statistical analysis, we used box-and-whisker diagrams,108 which present the distribution of data by a box and two whiskers. Minimum and maximum values are indicated by two horizontal lines. The first and third quartile of data, Q1 and Q3, define the bottom and top of the box where Q2 gives the position of the median. The interquartile range QR = Q3 − Q1 gives the vertical length of the box. The length of the whiskers is defined by Q1 − 1.5 QR (lower whisker) and Q3 + 1.5 QR (upper whisker). Any data point, which is lower than Q1 − q × QR and higher than Q3 + q × QR, is considered a mild outlier for q = 1.5 (black dots) and an extreme outlier for q = 3.0 (open dots).108 We have applied this analysis when at least more than 7 data points were available.

K−1d†n d nK−1d†n

(4)

where dn is now a row vector of matrix D. The local mode force constant kan is given by eq 5 kna = a†nKa n

and the local mode frequency (ωna)2 =

(5)

ωan

can be obtained from

Gnnkna 4π 2c 2

(6)

where element Gnn of matrix G defines the local mode mass.56 Before continuing it is useful to point out that the term local vibrational modes is sometimes used in a different context: Henry and co-workers94−96 use the term in connection with the (an)harmonic oscillator models to quantum mechanically calculate the overtones of XH stretching modes. The higher overtone modes (n = 5 or 6) for isolated XH groups are largely decoupled which justifies using the term local modes. Their frequencies correlate linearly with the Konkoli−Cremer local mode frequencies thus verifying their local mode character;97 however, they are only accessible for a few types of XH stretching modes, whereas the Konkoli−Cremer modes are generally defined and will be exclusively used in the following. The relative bond strength order (BSO) n of an OH bond is obtained by utilizing the extended Badger rule,97−99 according to which the BSO is related to the local stretching force constant ka by a power relationship, which is fully determined by two reference values and the requirement that for a zeroforce constant the BSO value becomes zero. Accordingly, the relationship n(OH) = a(ka)b can be derived from two suitable reference bonds. In this work, the constants a and b were determined for FH bonds using the frequencies of F−H (n = 1) and the D∞h-symmetrical [F···H···F]− anion (n = 0.5) as suitable references. Since the Badger and extended Badger rules predict for related XH bonds the same power relationship, the equation n(FH) = a(ka)b with a = 0.5402 and b = 0.2966 was also used for the OH bonds after shifting the single bond reference (corresponding to a n(OH) = 0.9653) by 0.0347 so that the OH bonds of H2O obtain the BSO value n = 1.00. In the MD simulations, 1000 water molecules in a PBC (periodic boundary condition) box were simulated by classical MD using the TIP5P100 force field in the NPT ensemble at 1.0 bar and 283, 308, 363, and 378 K where the simulations at 308 and 378 K were used as control calculations and therefore will not be discussed here in detail. The cutoff of nonbonded interactions was set to be 8 Å, and the Coulombic interactions were treated with the Ewald summation.101 The temperature was scaled by Langevin dynamics with the collision frequency γ being 1.0.102 The Berendsen bath coupling method103 was selected as a thermostat algorithm to control the pressure. The equations of the motion were integrated by the velocity Verlet algorithm104 with OH bond constrains.105,106 The time step was set to 1 fs (femtosecond), and the trajectories were collected for every 100 fs. The simulation time was 2 ns (nanoseconds) where the first ns was used for reaching the equilibrium. The trajectories of the second ns were used for the analysis. From the snapshots of the second ns, 1000 (H2O)1000 58


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Figure 2. Distribution of different H-bonds in the four 50-mers A (blue), B (orange), C (red), and D (green). The H-bonds are classified according to Figure 1 and ordered according to their frequency of appearance, which can be approximately described by an exponential dependence of the number N of H-bonds (P: position number). See text.

that for each water molecule 6 rather than 4 different H-bond arrangements become possible leading to a total of 62 = 36 Hbond types (see Figure S1 in the SI). The complete set of Hbond types will be discussed in Section 4. Finally, it has to be mentioned that similar notations on how to characterize different types of H-bonding in water clusters have been used in the literature (see Figure S1 of the SI for a relationship between these and the current notation).42,113−115 The numbering of atoms for the almost spherical structures of the four 50-mers is given in Figures S2−S5 of the SI. It is obvious that the water molecules located at the outside of the sphere have a smaller number of H-bonds than those water molecules positioned closer to the center of the 50-mer. In the four 50-mers A, B, C, and D, a total of 350 H-bonds (87,88,88,87) are found, which account for 15 of the 16 types of H-bonds shown in Figure 1. The 10-01 H-bond is topologically only possible in isolated cyclic water clusters as in (H2O)6 that we have used as a reference. In addition to the H-bond types of Figure 1, there are 20 H-bonds which were not found for the 50-mers but will be discussed in connection with the MD simulations. In Table S1 of the SI, all H-bonds identified for clusters A, B, C, and D are characterized by the R(H···O) distance, the local stretching force constant ka(H··· O), the associated local stretching frequency ωa(H···O), the BSO value n(H···O), and the O−H···O angle α. As one can see from the H-bond data (Table S1 in the SI, Figure 2), different types of H-bonding differ significantly in their number, where however their distribution is similar in the four water clusters. The distribution of the various types of Hbonding is nearly the same in the four 50-mers, which might be a result of their spherical form. The average number of Hbonds per water molecule, mav, is close to 3.5 in all cases (Table S1 in the SI). The most common H-bond is that of the 21-12 type (number of peripheral H-bonds: mp(D) = 3; mp(A) = 3; in short (3,3)), which accounts for 34.0% of all H-bonds followed by 11-12 ((2, 3); 14.3%), 21-11 ((3,2), 12.3%), and 11-11 Hbond ((2,2), 8.3%). The H-bond 11-02 is the least common

All vibrational mode and electron density calculations were carried out with the program package COLOGNE2016,109 whereas for the DFT and the GEBF calculations local versions of the program package Gaussian09110 were used. All MD simulations were performed with the AmberTools15 package.111 Difference densities were plotted with the program Multiwfn.112

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THE H-BONDING NETWORK IN WATER CLUSTERS The H-bond between a D- and an A-molecule can be characterized by the notation DcD(ia,jd)−AcA(ka,ld) where integers ia, jd, ka, and ld give the peripheral (external) Hbonds directly embedding the targeted H-bond, i.e. the acceptor (a) and donor (d) H-bonds of D and A, which for the D or A water molecule are normally equal to or smaller than 2. The superscripts cD and cA are the coordination numbers of O(D) and O(A), respectively, which vary between 3 and 4 in the case of the 50-mers. Hence, D4(20) − A3(02) (or in short (20-02)) denotes the H-bond with a D water molecule functioning as an acceptor for 2 external (=peripheral) H-bonds (its O atom coordinates with 4 H via normal or H-bonding; cD = 4) and an A water molecule that itself is functioning as a double donor for 2 other external H-bonds (cA = 3). This situation is sketched in the third row, second column of Figure 1. In the equilibrium geometry of the 50-mers, each D-molecule and each A-molecule are found to undergo one of the four possible interactions: 1) 10; 2) 11; 3) 20; 4) 21, which leads to a total of 16 = 42 different H-bond interactions such as 10-01, 10-02, 10-11, 10-12, ··· 21-12, where we have used as a reference the relatively strong 10-01 H-bond of cyclic (H2O)6. The largest number of peripheral H-bonds is realized for the Hbond D4(21) − A4(12), which has for both the D and A molecule 2 + 2 = 4 H-bonds and by this (ia + jd) + (ka + ld) = 3 + 3 = 6 peripheral H-bonds influencing the targeted D−A Hbond. Figure 1 contains additional information, which will be discussed below. In passing, we note that in liquid water, contrary to the equilibrium situation of a 50-mer, other relative weak H-bonds or no H-bonds at D and/or A are observed so 59


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Figure 3. Top: The bond strength order (BSO) n of the H-bond given as a function of the local H-bond stretching force constant ka(H···O). Hbonds are color-coded according to the notation of Figure 1. The average BSO value of each H-bond type is given by a vertical line where the length of the line indicates the range of BSO values found. Dashed vertical lines indicate those types of H-bonds for which only a few examples are observed. Bottom: The box-and-whisker diagram gives the statistical distribution of the BSO values for different types of H-bonds. For the definition of outliers (dots and circles), see Section 2.

type among the water molecules ((2,2) peripheral H-bonds) and accounts for just 1.4%. Other H-bonds such as 10-02 (0.3%), 10-11 (0.3%), 10-12 (2.6%), 11-01 (0.3%), 11-02 (1.4%), 11-22 (0.9%), 20-01 (0.3%), 20-11 (0.7%), 21-01 (2.6%), and 31-11 (1.0%) can only be found in one of the 50mers. In summary, there is an exponential decay of the statistical occurrence of specific H-bonds with the number and position of peripheral H-bonds (see below), which holds if one considers all four 50-mers together (Figure 2). Figure 3 provides an insight into which extent different Hbonds can be distinguished. The intrinsic strength of the Hbonds in the 50-mers as reflected by the BSO values varies by

50% from 0.225 to 0.425 (for comparison: 10-01 as in cyclic (H2O)6: n = 0.399). For each type of H-bond, the average BSO value is given by the crossing point of the curve n(ka) (top of Figure 3) and a vertical line, which defines the range of BSO values n for this particular type of H-bond by its length. In some cases, a definition of the range of n-values becomes meaningless because of the small number of H-bonds found for a particular type. Then, a dashed line of an arbitrary length of 0.04 BSO units is used. The representation at the top of Figure 3 is complemented by a statistical analysis of the different types of H-bonds among the 350 observed, which is given in the form 60


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D-side support this effect by pulling density from the O2(D) lp orbitals toward their OH bonds thus helping to increase the polarization of the OH bond of the D molecule (pull-ef fect, Figure 4). Hence, the polarization of the electron density distribution is in the direction of the red arrows shown for the 20-02 H-bond in Figures 1 and 4. At this point, it is appropriate to differentiate between physically based observables and the model quantities used in this work. Local mode frequencies and their associated force constants can be in principle measured.91 This is also true for the electron density, whereas NPA charges are model quantities connected with a special orbital model.81 Chemists explain Hbonding in terms of covalent, exchange, electrostatic, inductive, and dispersion interactions (see, e.g., Wang and co-workers116). Recently, Politzer and co-workers117−119 pointed out that according to the Hellmann−Feynman theorem120 noncovalent interactions are purely Coulombic in nature, and, accordingly, H-bonding might be described in this way. Although this is a valid view, often quantities such as NPA charges, charge transfer values, or charge delocalization energies provide a more detailed, model-based description of H-bonding. In this work, we will use the latter to describe covalent interactions. Difference densities can, as a result of their construction, reflect polarization effects although contributions from the changes in exchange repulsion, dispersion, etc. can also play a role. Apart from this, we will use an energy density based model that distinguishes just between covalent and electrostatic forces.86,87,121 Covalent versus Electrostatic H-Bonds. The covalent contribution of the H-bond can be considered as being dominated by a charge transfer from the lp(O1) orbital(s) to the σ*(O2−H) orbital of the D water (see bottom of Figure 5). The overlap between these orbitals will be maximal provided they are suitably oriented in line with an O2HO1 angle α close to 180° and an approach distance between (O2)H and O1 that is smaller than the sum of the van der Waals radii (1.2 + 1.52 Å122). Covalent contributions caused by charge transfer lead to H-bond stabilization as is reflected by the increase of the delocalization energy ΔE(del) with the BSO value (Figure 5, top), a more negative energy density at the H-bond critical point rb, and an accumulation of the electron density at this point.86,87,121 In the 50-mers investigated, the delocalization energies ΔE(del) vary from 8 to 32 kcal/mol, whereas the corresponding force constants vary from 0.29 to 0.41 mdyn/Å. It is striking that the ΔE(del) values fall into two groups (with the exception of that of type 10-11), which nicely correlate with the average BSO values (R2 = 1.00 and 0.97; Figure 5): The stronger Hbonds are presented by the upper line, which seems to combine those H-bond types with ia(D) + ld(A) > jd(D) + ka(A) due to ia = ld = 2 and jd + ka ≤ 1. Those H-bond types, which do not fulfill these criteria, are represented by the dashed lower line in Figure 5 (top). It seems that the strength of the various H-bond types is strongly influenced by the delocalization energy ΔE(del) where of course this is only valid for the 350 H-bonds of the four 50mers in their equilibrium geometries and the NPA approach used. One of the referees mentioned that for certain geometries two lp(O) of the same O can contribute to ΔE(del) of one Hbond. The analysis applied in this work revealed that for 79 out of 350 H-bonds a second lp(O) → σ*(OH) contribution larger than 3.0 kcal/mol (about one tenth of the first delocalization energy) was encountered, i.e. in 22.5% of all H-bonds, whereas

of a box-and-whisker diagram (bottom of Figure 3; for an explanation, see Section 2). The most stable H-bonds correspond to the 20-02 type (median: 0.411; average: 0.409; there is a linear relationship between median and average n values; R2 = 0.96; see the SI). They are followed by the 20-01 H-bonds (0.402), 10-02 (0.395), and 20-12 H-bonds (0.385; blue dots in Figure 3). The green dots of the 21-12 H-bonds, which one could expect as the strongest H-bonds (6 peripheral H-bonds) are quite frequent but belong to the weaker H-bonds because of their large variation from 0.269 to 0.421 with many of these bonds being in the low strength range as a result of geometrical constraints. If the H-bonds are ordered according to the sequence given in Figure 1, a decline of the intrinsic H-bond strength from 10-02 to 11-12, an incline to 20-01 and 20-02, and another decline to 21-11 is found. A H-bond turns out to be the strongest, which has just two external H-bonds on the D and two on the A side but which seems to guarantee both strong electrostatic and/or covalent interactions. This is is illustrated in Figure 1 for the 20-02 Hbond: Four peripheral water molecules polarize the electron density from A to D as is revealed by the difference density distribution Δρ(r) calculated for the 20-02 hexamer in Figure 4.

Figure 4. Diagrams show the difference density distribution Δρ(r) calculated for the 0.001 e/Bohr3 density surface of the hexamer defining the 20-02 type of H-bonding (see Section 2, for details). Blue contour lines indicate a depletion, red an increase of the electron density distribution because of polarization. Top: side view. Bottom: bird view.

This is defined in the way that the extra-effect of the four peripheral H2O molecules on the targeted 20-02 H-bond becomes visible. The two peripheral H2O at the A side polarize with their lp electrons the O−H bonds of A thus leading to an increase in the density close to O1(A) and in the 20-02 H-bond region (push-ef fect, Figure 4). The two peripheral H2O on the 61


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Figure 5. Top: The average delocalization energy ΔE(del) = ΔE(lp(O) → σ*(OH)) is plotted as a function of the average BSO n of the H-bond. For two different classes of H-bonds two correlation lines are obtained (not included: the 10-11 H-bond type). Bottom: The covalent contribution to the H-bond implies a delocalization of the lone pair (lp) electrons of atom O1 (green lobe of the orbital on the right) into the σ*(OH) antibonding orbital of the OH donor (green lobe of the orbital on the left). The energy stabilization caused by the charge transfer was determined by second order perturbation theory and is the basis for the delocalization energies of the upper diagram.

for 90% of all H-bonds the second lp contribution is between 0.05 and 3 kcal/mol. The perturbation theory analysis based on the NPA model also revealed that ΔE(del) contributions involving other than lp-orbitals are smaller than 0.1 kcal/mol, and therefore it is reasonable to use ΔE(del) for the description of the covalent contributions to the H-bond. An increase of the average number of H-bonds, m(AD), from 2 to 4 (Figure 1) does not necessarily weaken the H-bond but leads to a larger variation in the n-values because of stronger OH···O bending, the geometric limitations in the overlap between the lp(O1) and the σ*(O2H) orbitals, a less than optimal charge transfer, and lower covalent contributions to Hbonding, which might be only partly compensated by electrostatic contributions (see 21-12 in Figures 3, top, and 6, top). In Figure 6 (top), the geometrical conditions for H-bonding are compared. The box-and-whisker diagram gives the nonlinearity of the H-bond unit O2H···O1 as measured by the angle α. The analysis reveals that the stronger H-bonds (2002, 10-02, etc.) are more linear in agreement with the requirements for maximal overlap and charge transfer. The 21-12 H-bonds have the largest variation in α (almost 40°, Figure 6). Noteworthy is that the median values are in the range from 164 to 173° irrespective of the H-bond considered.

Since the nonlinearity of the H-bond arrangement is closely related to the covalent or electrostatic character of the H-bond, we show at the bottom of Figure 6 the statistical analysis of the energy density at the H-bond critical point, which should be negative for a dominant covalent bond according to the Cremer−Kraka criteria.86−88 This is qualitatively confirmed by the diagram, which gives an ”inverted” distribution of box-andwhisker units as compared to the diagrams in Figure 3. Accordingly, the 20-02 H-bond has the most negative energy density values, whereas the 21-12 H-bonds have at the same time the most positive energy density values and the largest variation of values. Variation in the Strength of a H-Bond. The explanation why the 20-02 H-bond is the strongest one has been based on Figures 1 and 4 (polarization of the density as indicated by the red arrows). Any deviation from this optimal arrangement leads to a weakening of the H-bond. Using the average BSO values of Figure 3, the following ordering according to decreasing Hbond strength results: 20-02 > 10-01 > 20-01 > 10-02 > 20-12 > 20-11 > 21-02 > 10-11 ≈ 21-01 > 10-12 > 11-02 > 11-01 > 21-11 > 21-12 > 11-12 > 11-11 (For the cases, with a maximum of five H-bonds per water molecule: 31-12 > 31-11 > 21-22 > 11-22, see Figure 3). This ordering can qualitatively be reproduced if one considers that any competition of the 62


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Figure 6. Top: The nonlinearity of the H-bond arrangement as measured by the angle α = O2H···O1 is analyzed in the form of a box-and-whisker diagram. For details of the box-and-whisker diagrams, see Section 2. Bottom: The box-and-whisker diagram gives the statistical distribution of the energy densities (given in Hartree/Bohr3) at the H-bond critical point for different types of H-bonds. Median values given at the top of the upper whisker are multiplied by 105.

weakening of −5 (two first order and one second order perturbation: −2 × 2−1 × 1 = −5) and thereby characterizes one of the weakest H-bonds (#14; in Figure 1. The ranking of each H-bond in terms of its average BSO is given by an encircled number: 1 gives the strongest and 16 the weakest Hbond; see also Figure 3). In this way, the ordering of most of the 16 H-bond types is correctly predicted (exceptions are 2011 and 10-11; the 10-01 value is added in position 3 using the BSO of the cyclic hexamer (H2O)6). For example, the 10-02 Hbond has a perturbation value of 10-00, i.e. only one of the outer H-bonds is missing thus yielding a second order weakening of the targeted D−A H-bond of −1 and position 3 in the list of strong H-bonds.

targeted H-bond with the second donor H-bond (as in 21-02, Figure 1) or with another external H-bond for the lp-density of the O(A) atom is a first order perturbation, whereas a change in the other peripheral H-bonds either on the D or A side can be considered as a second order perturbation (where perturbations on the D-side seem to have a somewhat larger effect than those on the A-side). Weighting first and second order perturbations qualitatively by −2 and −1 and using the short notation for the perturbation in the form |(ia, jd, ka, ld) − (20-02)| = (pi, pj, pk, pl) (p: perturbation), the negative blue numbers in Figure 1 are obtained (at the upper right of each H-bond arrangement). The qualitative comparison of a 21-11 H-bond with the 2002 reference leads to a perturbation 01-11, which implies a 63


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Figure 7. Top: The bond strength order (BSO) n of the O−H donor bond given as a function of the local O−H stretching force constant. Since each of these O−H-bonds is associated with a specific H-bond the former are identified and color-coded according to the associated H-bond (Figure 3). Bottom: A box-and-whisker diagram gives the statistical distribution of the BSO values of the various types of OH donor bonds leading to Hbonding in the four 50-mers investigated. For details of the box-and-whisker diagrams, see Section 2.

Polarization causes the energy of the σ*(O2H) orbital being lowered and that of the lp(O1) being raised thus effectively decreasing the energy gap between these orbitals and increasing the delocalization energy ΔE(del). In addition, electrostatic interactions can be maximized by the polarization effect. The analyses summarized in Figures 3 (BSO-values), 5 (delocalization energies ΔE(del)), and 6 (top: angles α; bottom: energy densities H(rb)) suggest that the covalent contributions to Hbonding are important for equilibrium geometries. This of course can be a consequence of (i) the finite size of the water clusters, (ii) the NPA model being used, and (iii) the exclusion of entropy effects in the current analysis.

Utilizing the perturbation indices given for each type of Hbond, a distortion relationship between the rows of Figure 1 can be obtained (purple numbers on the left of Figure 1). According to these values (row 1: −1; row 2: −3; row 3:0; row 4: −2), H-bonds 20-rs (r = 0, 1; s = 1, 2) are the most stable ones, followed by those of row 1, whereas the least stable ones are found in row 2. This ordering is directly related to the possible perturbations of the 20-02 arrangement, which reduce its push−pull effect and thereby its covalent character. The ordering of H-bonds by perturbation indices reveals that the polarization of the electron density of D and A as caused by the peripheral H2O molecules (Figure 4) is an important electronic effect for the intrinsic strength of the H-bond. 64


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Figure 8. Testing the relationship between D and A: Comparison of the local stretching force constants ka(H···O) and ka(O−H). The solid line is given by ka(O−H) = 7.938−1.211 × ka(H···O) − 9.917 × (ka(H···O))2 with R2 = 0.72 and σ = 0.30 mdyn/Å. For the H-bond notation, see Figure 1.

Figure 9. Testing the relationship between D and A: Comparison of the distance R(H···O) and R(O−H). The solid line is given by R(O−H) = 0.952−0.018 × log[R(H···O) − 1.552] with R2 = 0.93 and σ = 0.018 Å. For the H-bond notation, see Figure 1.

Donor−Acceptor Relationships. There are numerous investigations, which use vibrational spectroscopy to describe H-bonding123−136 and relate the strength of the O−H donor bond to the strength of the H-bond. 134,136,137 These investigations are mostly based on infrared spectroscopy as the weakening of the O−H donor bond in the case of Hbonding can be easily recorded by a red-shift of the OH stretching frequency. In a previous investigation, Freindorf and co-workers59 have shown that the expected relationship between the O−H donor bond and the H-bond is not fulfilled. However, this investigation included many different H-bond

donors. The current investigation is limited to D,A interactions between water molecules, and therefore a relationship between the O−H donor bond and the H-bond becomes more likely. As shown in Figure 7 (top), the BSO values of the O−H donor bonds present in the four 50-mers vary from 0.88 to 1.00 (water molecule without any H-bonding) and beyond this to 1.03 if also those OH bonds are included that are on the outside of the water cluster and therefore not involved in any H-bonding (denoted HO−H in Figure 7). Although the range of BSO values is only one-third of that of the H-bonds, the same number of bonding situations as found for the H-bonds 65


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Figure 10. Testing the Badger relationship: Comparison of the local H-bond stretching force constant ka(H···O) and the corresponding distance R(H···O). For the relationship ka = 0.648 R(O···H)2 + 3.181 R(O···H) + 3.878, R2 = 0.91 and σ = 0.023 mdyn/Å are calculated. For the H-bond notation, see Figure 1.

Figure 11. Testing the Badger relationship: Comparison of the local O−H stretching force constant ka(O−H) and the corresponding distance R(O− H). For the relationship ka(O−H) = −84.095 R(O−H) + 89.155, R2 = 0.98 and σ = 0.087 mdyn/Å result. For color code and notation, see Figures 1 and 7.

that the variation in the data points for the 20-02 donor bond is smaller than that for the corresponding H-bond, which again is a result of the fact that the first is primarily influenced by charge transfer and thereby a covalent weakening effect, whereas the latter is in addition influenced by electrostatic effects (for the calculated energy densities of the OH donor bonds, see the SI). Figure 7 also reveals that the various O−H donor bond types are much closer together with largely overlapping value ranges so that a differentiation on the basis of their average (Figure 7,

can be distinguished. Each is indicated in Figure 7 by a vertical solid or dashed line, which gives the average BSO value of a given O−H donor bond type that, for reasons of simplicity, is characterized by the H-bond it is engaged in. There is a qualitative relationship between O−H donor bonds and H-bonds insofar as the weakest O−H donor bonds (see 20-02 in Figure 7) are associated with the strongest Hbonds and vice versa. This is in line with the covalent character of the strong H-bonds, which implies charge transfer from O2(lp) into the σ*-orbital of the donor bond. Noteworthy is 66


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Table 1. Comparison of the Total Numbers and Percentages of All Types of Hydrogen Bonds for 1000 (H2O)1000 Clusters Determined along the MD Trajectories and Calculated with the TIP5P Force Field for an NPT Ensemble at 283 and 363 Ka H-bond type

N(283)

%

N(363)

%

20-02 20-01 10-01 10-02 20-12 group (1) 20-11 21-02 10-11 21-01 10-12 11-02 group (2) 11-01 21-12 21-11 11-12 11-11 group (3) groups (1+2+3) 00-00 00-01 00-02 00-10 00-11 00-12 01-00 01-01 01-02 01-10 01-11 01-12 10-00 10-10 11-00 11-10 20-00 20-10 21-00 21-10 group (4) 22-00 31-00 22-01 31-01 31-10 22-10 22-02 31-02 22-11 31-11 22-12 31-12 group (5) 00-22 00-13 01-22 10-22 01-13 10-13

22015 12337 7585 13163 91754 146854 37676 106723 24033 57788 57310 46680 330210 26433 471376 184060 210040 85691 977600 1454664 138 839 1399 309 2696 6310 351 2311 3912 869 7554 17432 1197 2630 3804 9641 1824 4060 8576 19580 95432 32 305 205 1960 465 93 242 3697 628 4541 1212 11712 25092 382 11 1230 2915 32 130

1.4 0.8 0.5 0.8 5.6 9.1 2.3 6.6 1.5 3.6 3.5 2.9 20.4 1.6 29.0 11.3 12.9 5.3 60.1 89.6

Different Ways of Hydrogen Bonding in Water - Why Does Warm Water

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